Graph theory

Prof Amir GevaEitan Netzer

G=(V,E)

•V – Vertex•E – Edges

Directed Graph

Undirected Graph

Weighted Graph

Representation

BFS

Shortest distance from vertex s to each vertex vWork on directed and undirected

First all are whiteDone with vertex blackElse grey

Time and space

BFSBFS(G,s) for each vertex u V[G]-{s} ∈

do color[u]<-white d[u]<-∞ π [u]<-NULL

color[s]<-gray d[s]<-0π [s]<-NULL Q<-{s} while Q≠ φ

do u<-head[Q] for each v Adj[u] ∈

do if color[v] = White then color[v]<- Grayd[v]<- d[u]+1 π [v]<- u Enqueue(Q,v)

Dequeue(Q) color[u]<- Black

BFS

DFS

Search all vertexes starting at vertex sNot have to be minimal

Time

Space

DFSDFS(G,s) for each vertex u V[G] ∈

do color[u]<-white π [u]<-NULL

time <- 0 for each vertex u V[G] ∈

do if color[u] = white then DFS-Visit(u)

DFS-Visit(u) color[u]<-Gray d[u] <- time <- time +1 for each vertex v Adj[u] ∈

do if color[v] =white then π [v]<-u

DFS-Visit(v) color[u] = Black f[u] <- time <- time +1

Improve weight

Relax(u,v,w) if d[v]>d[u] + w(u,v)

then d[v]<- d[u] + w(u,v) π [v]<- u

Dijkstra's algorithm

Find shortest path from vertex s to all other vertexes.Work on weighted directed and undirected.But non negative weights

List or array

Heap

DijkstraDijkstra(G,w,s) for each vertex u V[G] ∈

do d[u]<-∞ π [u]<-NULL

d[s]<-0 S <- φ Q<-V[G] while Q≠ φ

do u<- extract-min(Q) S<-S {u} ∪for each vertex v Adj[u] , Q∈

do Relax(u,v,w)

Negative example

Bellman–Ford algorithm

Find shortest path from vertex s to all other vertexes.Work on weighted directed can handle negative weights

Bellman–FordBellman-ford(G,w,s) for each vertex u V[G] ∈

do d[u]<-∞ π [u]<-NULL

d[s]<-0 for i <- 1 to |V(G)|-1

do for each edge (u,v) E[G] ∈do Relax(u,v,w)

for each edge (u,v) E[G] ∈do if d[v]>d[u]+w(u,v)

then return FALSE return TRUE